computer graphics3

MIT EECS 6.837, Cutler and Durand 1

Ray Casting

MIT EECS 6.837Frédo Durand and Barb Cutler

Some slides courtesy of Leonard McMillan

Courtesy of James Arvo and David Kirk. Used with permission.


Administrative• Assignment 1

– Due Wednesday September 17


Calendar• 1st quiz – Tuesday October 07th • 2nd quiz --Thursday Nov 20th• Week Dec 1-5 project presentation• Last day of class: December 9: best projects &

final report due


Questions?


Overview of the semester• Ray Tracing

– Quiz 1• Animation, modeling, IBMR

– Choice of final project• Rendering pipeline

– Quiz 2• Advanced topics


Overview of today

• Introduction

• Camera and ray generation

• Ray-plane intersection

• Ray-sphere intersection


Ray CastingFor every pixel

Construct a ray from the eye For every object in the scene

Find intersection with the ray Keep if closest


ShadingFor every pixel



Shade depending on light and normal vectore.g. diffuse shading:dot product N.La.k.a. Lambertian


Ray Tracing• Secondary rays (shadows, reflection, refraction)• In a couple of weeks


Ray representation?


Ray representation• Two vectors:

– Origin– Direction (normalized is better)

• Parametric line– P(t) = origin + t * direction

origindirection

P(t)


Ray Tracing• Original Ray-traced

image by Whitted

• Image computed using the Dali ray tracer by Henrik Wann Jensen

• Environment map by Paul Debevec

Image removed due to copyright considerations.

Image removed due to copyright considerations.


Ray castingFor every pixel


Find intersection with the rayKeep if closest

Shade depending on light and normal vector

Finding the intersection and normal is the central part of ray casting


Overview of today

• Introduction





CamerasFor every pixel


Find intersection with the ray

Keep if closest


Pinhole camera• Box with a tiny hole• Inverted image• Similar triangles

• Perfect image if hole infinitely small

• Pure geometric optics• No depth of field issue


Simplified pinhole camera• Eye-image pyramid (frustum)• Note that the distance/size of image are arbitrary


Camera description– Eye point e– Orthobasis u, v, w– Image distance s– Image rectangle

(u0, v0, u1, v1)

– Deduce c (lower left)– Deduce a and b– Screen coordinates in [0,1]*[0,1]– A point is then c + x a +y b

u

vw

ec

s

a

bx

y


Alternative perspective encoding• 4x4 matrix & viewing frustum• More about that next week


Orthographic camera• Parallel projection• No foreshortening• No vanishing point

perspective orthographic


Orthographic camera description


Orthographic camera description• Direction• Image center

• Image size• Up vector

size

size

upcenter


Orthographic ray generation• Direction is constant• Origin = center + (x-0.5)*size*up + (y-0.5)*size*horizontal

up

direction

horizontal

size(0,0)

(1,1)center

size


Other weird cameras• E.g. fish eye, omnimax, panorama


Overview of today

• Introduction








First we will study ray-plane intersection


Recall: Ray representation• Two vectors:

– Origin– Direction (normalized)

• Parametric line– P(t) = origin + t * direction

origindirection

P(t)


3D plane equation• Implicit plane equation

H(p) = Ax+By+Cz+D = 0• Gradient of H?

PH


3D plane equation• Implicit plane equation

H(p) = Ax+By+Cz+D = 0• Gradient of H?• Plane defined by

– P0(x,y,z,1)– n(A, B, C, 1)

P0PH


Explicit vs. implicit?• Plane equation is implicit

– Solution of an equation– Does not tell us how to generate a point on the plane– Tells us how to check that a point is on the plane

• Ray equation is explicit– Parametric– How to generate points– Harder to verify that a point is on the ray


Plane-point distance• Plane Hp=0• If n is normalized

d=HP• Signed distance!

P’

P0

P

H


Explicit vs. implicit?• Plane equation is implicit

– Solution of an equation– Does not tell us how to generate a point on the plane– Tells us how to check that a point is on the plane

• Ray equation is explicit– Parametric– How to generate points– Harder to verify that a point is on the ray

• Exercise: explicit plane and implicit ray

Line-plane intersection• Insert explicit equation of line into

implicit equation of plane

origindirection

P(t)


Additional house keeping• Verify that intersection is closer than previous• Verify that it is in the allowed range

(in particular not behind the camera, t<0)



Normal• For shading (recall, diffuse: dot product between

light and normal)• Simply the normal to the plane

origindirection

P(t)normal


Overview of today

• Introduction




Sphere equation


• Sphere equation (implicit): ||P––2 = r2

• (assume centered at origin, easy to translate)

Ray-Sphere Intersection


• Sphere equation (implicit): ||P––2 = r2

• Ray equation (explicit): P(t) = R+tDwith ||D|| = 1

• Intersection means both are satisfied



• This is just a quadratic at2 + bt + c = 0, where

a = 1

b = 2D.R

c = R.R – r2

• With discriminant

• and solutions



• Discriminant• Solutions• Three cases, depending on sign of b2 –4ac• Which root (t+ or t should you choose?

– Closest positive! (usually t-)



• So easy that all ray-tracing images have spheres!


Geometric ray-sphere intersection• Try to shortcut (easy reject)• e.g.: if the ray is facing away from the sphere• Geometric considerations can help

• In general, early reject is important

R

r

O

D


Geometric ray-sphere intersection• What geometric information is important?

– Inside/outside– Closest point– Direction

Rr

O

D


Geometric ray-sphere intersection• Find if the ray’s origin is outside the sphere

– R2>r2

– If inside, it intersects – If on the sphere, it does not intersect (avoid degeneracy)

Rr

O

D


Geometric ray-sphere intersection• Find if the ray’s origin is outside the sphere• Find the closest point to the sphere center

– tP=RO.D– If tP<0, no hit

Rr

O

DtPP



– If tP<0, no hit

• Else find squared distance d2

– Pythagoras: d2=R2-tP2

– …

Rr

O

DP tP

d

if d2> r2 no hit



– If tP<0, no hit• Else find squared distance d2

– if d2 > r2 no hit

• If outside t = tP-t’– t’2+d2=r2

• If inside t = tP+t’

R

rO

DP tP

d t

t’


Geometric vs. algebraic• Algebraic was more simple

(and more generic)• Geometric is more efficient

– Timely tests– In particular for outside and pointing away


Normal• Simply Q/||Q||

R

rO

D

t

Qnormal


Precision• What happens when

– Origin is on an object?– Grazing rays?

• Problem with floating-point approximation


The evil ε• In ray tracing, do NOT report intersection for

rays starting at the surface (no false positive)– Because secondary rays– Requires epsilons


The evil ε: a hint of nightmare• Edges in triangle meshes

– Must report intersection (otherwise not watertight)– No false negative


Assignment 1• Write a basic ray caster

– Orthographic camera– Spheres– Display: constant color and distance

• We provide – Ray – Hit– Parsing– And linear algebra, image


Object-oriented design• We want to be able to add primitives easily

– Inheritance and virtual methods• Even the scene is derived from Object3D!

Object3Dbool intersect(Ray, Hit, tmin)

Planebool intersect(Ray, Hit, tmin)

Spherebool intersect(Ray, Hit, tmin)

Trianglebool intersect(Ray, Hit, tmin)

Groupbool intersect(Ray, Hit, tmin)


Ray//////////class Ray {//////////

Ray () {}Ray (const Vec3f &dir, const Vec3f &orig){_dir =dir; _orig=orig;}Ray (const Ray& r) {*this=r;}

const Vec3f &origin() {return _orig;}const Vec3f &direction() {return &dir;}Vec3f pointAtParameter (float t) {return _orig+t*_dir;}

private:Vec3f _dir;Vec3f _orig;

};


Hit• Store intersection point & various information

/////////class Hit {

/////////

float _t;

Vec3f _color;//Material *_material;//Vec3f _normal;

};


Tasks• Abstract Object3D• Sphere and intersection• Scene class• Abstract camera and derive Orthographic• Main function


Thursday: More Ray Casting• Other primitives

– Boxes– Triangles– IFS?

• Antialiasing

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